Renormalization, rigidity, and universality in bifurcation theory.

Title: Renormalization, rigidity, and universality in bifurcation theory.
Identifier: AAI9605604
identifier: 9605604
Creator: Hu, Jun.
Contributor: Adviser: Dennis P. Sullivan
Date: 1995
Language: English
Publisher: City University of New York.
Subject: Mathematics
Abstract: We aim to give a possible explanation why smooth one-parameter families of threadlike mappings, which pass from simple dynamical systems to chaotic dynamical systems, generically exhibit the asymptotic geometric rigidity in period-doubling bifurcation: there are infinite sequences {dollar}\{lcub}\mu\sb{lcub}n{rcub}\{rcub}{dollar} of parameter values such that at {dollar}\{lcub}\mu\sb{lcub}n{rcub}\{rcub}{dollar} there is a loss of a stable periodic trajectory of period 2{dollar}\sp{lcub}n{rcub}{dollar} and a rise of a stable periodic trajectory of period 2{dollar}\sp{lcub}n+1{rcub},{dollar} and the ratios of adjacent parameter changes tend to a constant 4.6692{dollar}\cdot\cdot\cdot.{dollar} We show that this statement is true in the space of smooth one-dimensional maps with finitely many critical points.;We also show that in the space of real-coefficient one-variable polynomials of degree {dollar}d\geq{dollar} 1, the set of the maps at the accumulations of period-doubling bifurcations forms the boundary of chaos, where the chaos is the set of the maps with positive topological entropy.
Type: dissertation
Source: PQT Legacy CUNY.xlsx
degree: Ph.D.